Shannon-Hartley theorem can accurately calculate the channel capacity when the signal observation time is infinite. However, the calculation of finite-time capacity, which remains unknown, is essential for guiding the design of practical communication systems. In this paper, we investigate the capacity between two correlated Gaussian processes within a finite-time observation window. We first derive the finite-time capacity by providing a limit expression. Then we numerically compute the maximum transmission rate within a single finite-time window. We reveal that the number of bits transmitted per second within the finite-time window can exceed the classical Shannon capacity, which is called as the Exceed-Shannon phenomenon. Furthermore, we derive a finite-time capacity formula under a typical signal autocorrelation case by utilizing the Mercer expansion of trace class operators, and reveal the connection between the finite-time capacity problem and the operator theory. Finally, we analytically prove the existence of the Exceed-Shannon phenomenon in this typical case, and demonstrate the achievability of the finite-time capacity and its compatibility with the classical Shannon capacity.
翻译：当信号观测时间无限时,Shannon-Hartley 光学理论可以准确计算频道容量。 然而,计算有限时间容量对于指导实用通信系统的设计至关重要,但这一计算仍然未知。 在本文中,我们调查了一个有限时间观测窗口内两个相关高斯进程之间的容量。 我们首先通过提供一定的表达式来得出有限时间容量。 然后我们用数字计算一个单一有限时间窗口中的最大传输率。 我们发现,在限定时间窗口中每秒传输的位数可能超过典型的香农容量,即所谓的Exceed-Shannon 现象。 此外,我们通过利用追踪类操作员的Mercer扩展,在典型信号自动变异的情况下得出了一个固定时间容量公式,并揭示了有限时间容量问题与操作员理论之间的联系。 最后,我们分析地证明,在这个典型案例中存在超时间-Shannon现象,并表明有限时间容量及其与典型香农能力的兼容性。
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